50 Years of First-Passage Percolation

Abstract

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2 + log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

Bigeodesics in First-Passage Percolation

“…The limit shape referred to above is the set B ⊂ R 2 given by B = {z : g(z) ≤ 1}, and g is the asymptotic norm for the FPP model (see [ 2 . B is also characterized by the "shape theorem," (see [3,Theorem 2.6] and [6]) which says that given ǫ > 0, one has P ((1 − ǫ)B ⊂ B(t)/t ⊂ (1 + ǫ)B for all large t) = 1, where B(t)/t is the set {z/t : z ∈ B(t)}. There is no simple condition known to guarantee that the limit shape under condition A2' is bounded.…”

“…There is no simple condition known to guarantee that the limit shape under condition A2' is bounded. A1' and A2' are sufficient to ensure that between each x, y ∈ Z 2 , there is a unique geodesic Γ(x, y) (see [3,Section 4.1]).…”

Book Review: Noise sensitivity of Boolean functions and percolation

“…Thus we get |γ| ≤ b a v 1 . This means that we need to only look at a finite box around the origin in order to find d ω (0, v).…”

“…First passage percolation (FPP) is a model for a random metric on a graph; see [1] for a thorough review of FPP. Consider the edge set E of Z d .…”

Book Review: 50 years of first-passage percolation

“…. Here we will concentrate on the former, referring the reader to [5] for further detail and a discussion on the nonrandom error.…”